Gaifman property from relative categoricity

Determine whether relative categoricity over the distinguished predicate P implies the Gaifman property, namely, whether every model of the induced theory of P occurs as the P-part of some model of the complete theory T.

Background

The paper distinguishes the Gaifman property from relative categoricity over the distinguished unary predicate P. The Gaifman property requires every model of the induced theory of P to arise as the P-part of a model of T, whereas relative categoricity requires any two models with the same P-part to be isomorphic by an isomorphism fixing that part pointwise. The unresolved question asks whether the latter condition guarantees the former.

References

Gaifman's original question asks whether relative categoricity implies the Gaifman property p.~30; see also the modern formulation in Definition~2.2. The question remains open p.~1.

— Counterexamples to the Generalized Gaifman Conjecture  (2608.16099 - Zhang, 17 Aug 2026) in Section 1, Introduction